Answer
$x =\dfrac{\ln{\pi}}{1+\ln{\pi}}\approx 0.534$
Work Step by Step
To find the value of $x$, we take the $\log$ of both sides and then isolate $x$ as follows:
$$\ln (\pi^{1-x})=\ln (e^{x})$$
Recall :
$\log m^n= n \log m$
Use the rule above to obtain:
$(1-x)(\ln\pi)=x \\
\ln{\pi}-x\cdot \ln{\pi}=x\\
\ln\pi=x+x\cdot\ln\pi$
Factor out $x$ on the right side, then solve for $x$ to obtain:
$\ln\pi=x(1+\ln\pi)\\
\dfrac{\ln\pi}{1+\ln\pi}=x\\
x \approx 0.534$
Thus, our answer is: $x =\dfrac{\ln{\pi}}{1+\ln{\pi}}\approx 0.534$
